In many real world optimization problems it is essential or desirable to determine the global minimum of the objective function. The subject of this dissertation is a new class of methods that tackle such problems. In particular, we have in mind problems where function evaluations are expensive and no additional information is available. The methods employ radial basis functions that have been proved to be useful for interpolation problems. Examples include thin plate splines and multiquadrics. Specifically, in each iteration, radial basis function interpolation is used to define a utility function. A maximizer of this function is chosen to be the next point where the objective function is evaluated. Relations to similar optimization methods are established, and a general framework is presented that combines these methods and our methods. A large part of the dissertation is devoted to the convergence theory. We show that convergence can be achieved for most types of basis functions without further assumptions on the objective function. For other types, however, a similar results could not be obtained. This is due to the properties of the so-called native space that is associated with a basis function. In particular, it is of interest whether this space contains sufficiently smooth functions with compact support. In order to address this question, we present two approaches. First, we establish a characterization of the native space in terms of generalized Fourier transforms. For many types, for example thin plate splines, this helps to derive conditions on the smoothness of a function that guarantee that it is in the native space. For other types, for example multiquadrics, however, we show that the native space does not contain a nonzero function with compact support. The second approach we present gives slightly weaker results, but it employs some new theory using interpolation on an infinite regular grid.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:599804 |
| Date | January 2002 |
| Creators | Gutmann, H.-M. |
| Publisher | University of Cambridge |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
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